New properties of the Fucik spectrum

In this Note we present some results on the Fucik spectrum for the Laplace operator, that give new information on its structure. In particular, these results show that, if Omega is a bounded domain of R^N with N > 1, then the Fucik spectrum has infinitely many curves asymptotic to the lines {lambda(1)} x R and R x {lambda(1)}, where lambda(1) denotes the first eigenvalue of the operator -Delta in H_0^1(Omega). Notice that the situation is quite different in the case N = 1; in fact, in this case, the Fucik spectrum may be obtained by direct computation and one can verify that it includes only two curves asymptotic to these lines. The method we use for the proof is completely variational.